Independent/Dependent Events

Two events are
independent
if the result of the second event is not affected by the result of the first event. If
A
and
B
are independent events, the
probability
of both events occurring is the product of the probabilities of the individual events.

P
(
A
and
B
)
=
P
(
A
)
⋅
P
(
B
)

Example 1:

A box contains
4
red marbles,
3
green marbles and
2
blue marbles. One marble is removed from the box and then replaced. Another marble is drawn from the box. What is the probability that the first marble is blue and the second marble is green?

Because the first marble is replaced, the size of the sample space (
9
) does not change from the first drawing to the second so the events are independent.

Two events are
dependent
if the result of the first event affects the outcome of the second event so that the probability is changed. In the above example, if the first marble is not replaced, the sample space for the second event changes and so the events are dependent. The probability of both events occurring is the product of the probabilities of the individual events:

P
(
A
and
B
)
=
P
(
A
)
⋅
P
(
B
)

Example 2:

A box contains
4
red marbles,
3
green marbles and
2
blue marbles. One marble is removed from the box and it is not replaced. Another marble is drawn from the box. What is the probability that the first marble is blue and the second marble is green?

Because the first marble is not replaced, the size of the sample space for the first marble (
9
) is changed for the second marble (
8
) so the events are dependent.